The Old Testament repeatedly identifies the LORD as Israel's Savior, while human agents can also function as deliverers. Luke's combination of titles must therefore be read cumulatively.
The heavenly messenger identifies the newborn Jesus as Savior, Messiah, and Lord within a narrative already connecting Jesus with the promised divine visitation.
Luke does not depend on one title alone. The cluster of titles and the larger infancy narrative make this a strong declaration of Jesus' elevated identity.
The shepherds go to see the child and later glorify and praise God.
Their response confirms the angelic announcement as divine revelation rather than human speculation.